A251660 Table of coefficients in functions R(n,x) defined by R(n,x) = exp( n*x*G(n,x)^(n-1) ) / G(n,x)^(n-1) where G(n,x) = 1 + x*G(n,x)^n, for rows n>=1.
1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 8, 1, 1, 1, 4, 21, 56, 1, 1, 1, 5, 40, 261, 592, 1, 1, 1, 6, 65, 712, 4833, 8512, 1, 1, 1, 7, 96, 1505, 18784, 120303, 155584, 1, 1, 1, 8, 133, 2736, 51505, 663424, 3778029, 3456896, 1, 1, 1, 9, 176, 4501, 115056, 2354725, 29480896, 143531433, 90501632, 1
Offset: 1
Examples
This table begins: n=1: [1, 1, 1, 1, 1, 1, 1, 1, ...]; n=2: [1, 1, 2, 8, 56, 592, 8512, 155584, ...]; n=3: [1, 1, 3, 21, 261, 4833, 120303, 3778029, ...]; n=4: [1, 1, 4, 40, 712, 18784, 663424, 29480896, ...]; n=5: [1, 1, 5, 65, 1505, 51505, 2354725, 135258625, ...]; n=6: [1, 1, 6, 96, 2736, 115056, 6455376, 454666176, ...]; n=7: [1, 1, 7, 133, 4501, 224497, 14926387, 1245099709, ...]; n=8: [1, 1, 8, 176, 6896, 397888, 30584128, 2948178304, ...]; n=9: [1, 1, 9, 225, 10017, 656289, 57255849, 6262226721, ...]; n=10:[1, 1, 10, 280, 13960, 1023760, 99935200, 12226859200, ...]; ... where e.g.f. of row n equals: exp( n*x*G(n,x)^(n-1) ) / G(n,x)^(n-1). Related table of coefficients in G(n,x) = 1 + x*G(n,x)^n begins: n=1: [1, 1, 1, 1, 1, 1, 1, 1, ...]; n=2: [1, 1, 2, 5, 14, 42, 132, 429, ...]; n=3: [1, 1, 3, 12, 55, 273, 1428, 7752, ...]; n=4: [1, 1, 4, 22, 140, 969, 7084, 53820, ...]; n=5: [1, 1, 5, 35, 285, 2530, 23751, 231880, ...]; n=6: [1, 1, 6, 51, 506, 5481, 62832, 749398, ...]; n=7: [1, 1, 7, 70, 819, 10472, 141778 , 1997688, ...]; n=8: [1, 1, 8, 92, 1240, 18278, 285384, 4638348, ...]; n=9: [1, 1, 9, 117, 1785, 29799, 527085, 9706503, ...]; n=10:[1, 1, 10, 145, 2470, 46060, 910252, 18730855, ...]; ...
Crossrefs
Programs
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PARI
{T(n,k)=local(G=1); for(i=0, k, G=1+x*G^n +x*O(x^k)); k!*polcoeff(exp(n*x*G^(n-1))/G^(n-1), k)} /* Print as a rectangular table */ for(n=1, 10, for(k=0,10, print1(T(n,k), ", "));print("")) /* Print as a flattened table */ for(n=0, 12, for(k=0,n, print1(T(n-k+1,k), ", "));) /* Print the Related table of functions G(n,x) = 1 + x*G(n,x)^n */ {R(n,k)=local(G=1); for(i=0, k, G=1+x*G^n +x*O(x^k)); polcoeff(G, k)} for(n=1, 10, for(k=0,10, print1(R(n,k), ", "));print("")) -
PARI
/* Binomial sum formula for term T(n,k) */ {T(n,k) = if(k<=1,1,sum(j=0,k, n^j * k!/j! * binomial(n*(k-1)-j, k-j) * (j-1)/(k-1)))} for(n=1, 10, for(k=0, 10, print1(T(n, k), ", ")); print(""))
Formula
E.g.f. of row n, R(n,x), for n>=1, satisfies:
(1) [x^k/k!] R(n,x)^(k+1) = n^(k-1) * (n+k) * (k+1)^(k-2) for k>=0.
(2) R(n,x) = exp( n*x*G(n,x)^(n-1) ) / G(n,x)^(n-1), where G(n,x) = 1 + x*G(n,x)^n.
(3) R'(n,x)/R(n,x) = G(n,x)^(n-1), where G(n,x) = 1 + x*G(n,x)^n.
T(n,k) = Sum_{j=0..k} n^j * k!/j! * binomial(n*(k-1)-j, k-j) * (j-1)/(k-1) for k>1, n>=1.